Mathematik-Online-Lexikon: Spezielle Reihen

$\displaystyle \sum_{k=0}^\infty aq^k$	$\displaystyle = a + aq + aq^2 + \cdots = \frac{a}{1-q}$ , für $\displaystyle \vert q\vert< 1$
$\displaystyle \sum_{k=1}^\infty \frac{1}{k}$	$\displaystyle = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty$
$\displaystyle \sum_{k=1}^\infty (-1)^{k-1} \frac{1}{k}$	$\displaystyle = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \pm \cdots = \ln{2}$
$\displaystyle \sum_{k=1}^\infty \frac{1}{k^2}$	$\displaystyle = 1 +\frac{1}{2^2} +\frac{1}{3^2} + \cdots = \frac{\pi^2}{6}$
$\displaystyle \sum_{k=1}^\infty (-1)^{k-1}\frac{1}{k^2}$	$\displaystyle = 1 - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} \pm \cdots = \frac{\pi^2}{12}$
$\displaystyle \sum_{k=0}^\infty \frac{1}{k!}$	$\displaystyle = 1+ \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots = e$
$\displaystyle \sum_{k=0}^\infty (-1)^k \frac{1}{k!}$	$\displaystyle = 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} \pm \cdots = \frac{1}{e}$
$\displaystyle \sum_{k=1}^\infty \frac{1}{k\cdot 2^k}$	$\displaystyle = \frac{1}{2} + \frac{1}{2\cdot 2^2} + \frac{1}{3\cdot 2^3} + \cdots = \ln 2$

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	Mathematik-Online-Lexikon:
	Spezielle Reihen